I have asked for a Buddhist name from the temple, the Geshe la ask the H.H Dalai Lama office from India to give me a name. And I received a card from the office with the name of Tenzin Nangzey.

Last Saturday I attended a ceremony with Rinpoche, as the Geshe la from the temple suggest me to come to celebrate my new name with new clothes and all. But in the ceremony the Rinpoche, he gave me the name of Tenzin Sandrup.

Both names I don't know the exact meaning. Would some one be able to help translate the meaning of the two names above?
Which one should I choose? I guess I will wait to understand the meanings first.

You will find that over the course of time that you will accumulate a number of Dharma names. Don't stress about it. What you do is WAY more important than what you are called!

"My religion is not deceiving myself."Jetsun Milarepa 1052-1135 CE

"Butchers, prostitutes, those guilty of the five most heinous crimes, outcasts, the underprivileged: all are utterly the substance of existence and nothing other than total bliss."The Supreme Source - The Kunjed Gyalpo
The Fundamental Tantra of Dzogchen Semde